Question

The ratio of the specific heat capacities of a gas is 1.5. When the gas undergoes an adiabatic process, its volume is doubled and pressure becomes \( P_1 \). When the gas undergoes isothermal process, its volume is doubled and pressure becomes \( P_2 \). If \( P_1 = P_2 \), the ratio of the initial pressures of the gas when it undergoes adiabatic and isothermal processes is:

• A. \( \sqrt{3}: \sqrt{2} \)
• B. \( 1: 1 \)
• C. \( \sqrt{3}:1 \)
• D. \( \sqrt{2}:1 \) \bigskip

Hint:

In an isothermal process, the pressure and volume follow Boyle's Law: \[ P V = \text{constant} \] whereas in an adiabatic process, the relation follows Poisson's equation: \[ P V^\gamma = \text{constant} \] For a given ratio of heat capacities \( \gamma = 1.5 \), using these relations, we can derive the pressure ratio as \( 2:1 \). Understanding these fundamental gas laws helps in solving thermodynamics problems efficiently.

Answer 1

The Correct Option is D

Step 1: Understanding the Given Conditions - For an isothermal process, the pressure-volume relation is given by Boyle’s Law: \[ P V = \text{constant} \] Thus, if volume is doubled, the new pressure is: \[ P_2 = \frac{P_{\text{initial}}}{2} \] - For an adiabatic process, the pressure-volume relation is given by Poisson’s equation: \[ P V^\gamma = \text{constant} \] where \( \gamma \) is the adiabatic index, given as 1.5. 

Step 2: Applying the Adiabatic Equation For an adiabatic process: \[ P_{\text{initial}} V^\gamma = P_1 (2V)^\gamma \] Rearranging: \[ P_1 = P_{\text{initial}} \times (2)^{-\gamma} \] Substituting \( \gamma = 1.5 \): \[ P_1 = P_{\text{initial}} \times (2)^{-1.5} = \frac{P_{\text{initial}}}{2^{1.5}} \] Since \( 2^{1.5} = \sqrt{8} = 2\sqrt{2} \), we get: \[ P_1 = \frac{P_{\text{initial}}}{2\sqrt{2}} \] \

Step 3: Equating \( P_1 \) and \( P_2 \) Given that \( P_1 = P_2 \), we equate the expressions: \[ \frac{P_{\text{initial}}}{2\sqrt{2}} = \frac{P_{\text{initial}}}{2} \] Solving for the pressure ratio: \[ \frac{P_{\text{initial}}}{P_{\text{initial (iso)}}} = \sqrt{2}:1 \] Thus, the correct ratio is: \[ \mathbf{\sqrt{2}:1} \]